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Questions and Answers
What is the inverse operation of addition?
What is the inverse operation of addition?
The range of a function can also refer to its outputs.
The range of a function can also refer to its outputs.
True (A)
What is the domain of a function related to?
What is the domain of a function related to?
Input values
The inverse of multiplication is __________.
The inverse of multiplication is __________.
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Match the following terms with their definitions:
Match the following terms with their definitions:
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If a function f(x) has a domain of all real numbers, which of the following is true?
If a function f(x) has a domain of all real numbers, which of the following is true?
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All functions must be linear.
All functions must be linear.
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What must an element of a domain have for a function to be defined?
What must an element of a domain have for a function to be defined?
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For a function to be considered valid, it is necessary that each input must have __________ values.
For a function to be considered valid, it is necessary that each input must have __________ values.
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Which of the following statements is true regarding elements of a domain?
Which of the following statements is true regarding elements of a domain?
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What condition must a function meet to be considered one-to-one?
What condition must a function meet to be considered one-to-one?
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A function that is not one-to-one may still have an inverse.
A function that is not one-to-one may still have an inverse.
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What does the notation (5,4) represent in the context of coordinates?
What does the notation (5,4) represent in the context of coordinates?
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A function that is _____ must have a unique output for every input.
A function that is _____ must have a unique output for every input.
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Which of the following represents a valid function output for the input (5,10)?
Which of the following represents a valid function output for the input (5,10)?
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A function can have the same output for different inputs.
A function can have the same output for different inputs.
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What happens if a function has multiple outputs for the same input?
What happens if a function has multiple outputs for the same input?
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The range of a function is determined by its _____ values.
The range of a function is determined by its _____ values.
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What is the purpose of the vertical line test?
What is the purpose of the vertical line test?
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Study Notes
Introduction to Functions
- A function is a relationship between inputs where each input is related to exactly one output.
- Functions have a domain (set of inputs) and a codomain/range (set of possible outputs).
- Functions are typically denoted as f(x), where x represents the input.
- Examples of functions include: f(x) = sin x, f(x) = x² + 3, f(x) = 1/x, and f(x) = 2x + 3.
Types of Functions
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One-to-one function (Injective Function): Each input maps to a unique output, and no two inputs map to the same output.
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Many-to-one function: Multiple inputs can map to the same output.
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Onto function (Surjective Function): Every element in the codomain is mapped to by at least one element in the domain.
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Bijective Function: A function that is both one-to-one and onto. Invertible functions are bijective.
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Other function types mentioned are: Linear, Quadratic, Rational, Algebraic, Cubic, Modulus, Signum, Greatest Integer, Fractional Part, Even and Odd, and Periodic.
Invertible Functions
- The inverse of a function f, denoted as f⁻¹(x), reverses the relationship, mapping outputs back to inputs.
- A function is invertible if and only if it is bijective (one-to-one and onto).
- For a function to have an inverse, each output must correspond to a unique input.
- The inverse of adding is subtracting, and the inverse of multiplying is dividing.
Graphical Representation
- The domain of a function is the set of inputs, and the range is the set of outputs.
- The domain of f⁻¹(x) is the range of f(x), and the range of f⁻¹(x) is the domain of f(x).
- Visual representation aids understanding of function relationships and inverses.
Reflexive Property of Invertible Functions
- If a point (a, b) is on the graph of a function f, then the point (b, a) must be on the graph of the inverse function f⁻¹.
- This property demonstrates the reflection of the graph of a function over the line y = x when finding the inverse.
Conditions for Invertibility
- A function is invertible if it's both one-to-one and onto over its entire domain.
- A function may be one-to-one in certain parts of its domain but not over the entire domain.
Horizontal Line Test
- Used to determine if a function is invertible (one-to-one).
- If any horizontal line intersects the graph of the function more than once, the function is not one-to-one and does not have an inverse.
Demonstration of an Invertible Function
- Graphing y = 2x and y = x/2, illustrates the relationship between a function and its inverse.
- The inverse graph is a reflection of the original graph across the line y = x.
Properties of the Inverse Graph
- The inverse of a function is found by exchanging x and y coordinates in its graph.
- Graph of the inverse of f is a reflection across the y = x line.
- Example in page 9 details coordinates shifting in this relation.
Conclusion regarding invertibility
- Functions must be one-to-one to have an inverse.
- Exchanging input and output rows is how to find the inverse of tabular functions.
- Many "toolkit functions" have inverses.
- Graphically, an inverse is a reflection across the line y = x.
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Description
This quiz covers the fundamentals of functions, including their definitions, types, and notations. Students will explore one-to-one, many-to-one, onto, and bijective functions. Enhance your understanding of how functions relate inputs to outputs with practical examples.
